Abstract. In this survey we discuss a common feature of some classical and recent results in number theory, graph theory, etc. We try to point out the fascinating relationship between the theory of uniformly distributed sequences and Ramsey theory by formulating the main results in both fields as statements about certain irregularities of partitions. Our approach leads to some new problems as well.

INTRODUCTION

In 1916 Hermann Weyl published his classical paper entitled “Ober die Gleichverteilung von Zahlen mod Eins”. This was intended to furnish a deeper understanding of the results in diophantine approximation and to generalize some basic results in this field. The theory of uniformly distributed sequences has originated with this paper. In the last decades this subject has developed into an elaborate theory related to number theory, geometry, probability theory, ergodic theory, etc.

Curiously enough, Issai Schur's paper entitled “Ober die Kongruenz xn+yn≡zn (mod p)” appeared in the very same year. He proved that if the positive integers are finitely colored, then there exist x, y, z having the same color so that x+y=z. Though Ramsey theory has various germs, Schur's theorem can be regarded as the first Ramsey-type theorem. Now literally the same applies to Ramsey theory as to the theory of uniform distribution:

In the last decades Ramsey theory became an elaborate theory related to number theory, geometry, probability theory, ergodic theory, etc.

This paper is dedicated to the memory of J. Howard Redfield.

INTRODUCTION

Several years ago in a lecture at a British Mathematical Colloquium splinter group (Lancaster meeting, 1978) Dr. E.K. Lloyd mentioned that he had reason to believe that J. Howard Redfield had written a second (but unpublished) paper. As is well-known Redfield's first paper [17], published in 1927, was badly neglected although at least published. Attention as far as one knows was first drawn to it by Littlewood [13] in 1950. The paper was first publicized by Harary in 1960 [10]. It seemed that Redfield had anticipated most of the later discoveries in the theory of unlabelled enumeration such as Pólya's Hauptsatz, Read's Superposition Theorem, the counting of nonisomorphic graphs and the counting of self-complementary subsets of a set with respect to a group. At the end of 1981 I received an exciting letter from Lloyd stating that this second paper [18] had indeed been found. Apparently the paper had been submitted for publication in the American Journal of Mathematics on October 19th, 1940 and was rejected by the editors in a brief letter of January 7th, 1941. Harary and Robinson [12] have written a brief account of the circumstances leading up to the discovery of Redfield's second paper and happily a special edition of the Journal of Graph Theory is to be published entirely dedicated to Redfield.

For there is no man can Write fo warily, but that he may fometime give Opportunity of Cavilling, to thofe who feek it. John Wallis, A treatife of Algebra, 1685.

From rather modest beginnings the British Combinatorial Conference has grown into an established biennial international gathering. A successful format for the Conference has been established whereby several distinguished mathematicians are each invited to give a survey lecture at the Conference and to write a paper for the Conference Volume, which is published in time for the start of the meeting. The present volume contains eight of the nine invited papers for the Ninth Conference held at the University of Southampton, 11-15 July 1983.

Between them the papers cover a broad range of combinatorics. The all-pervading subject of graph theory appears in a number of the papers. It is the central feature of the one by J.C. BERMOND and his co-authors in which they survey those results concerning diameter and connectivity in graphs and hypergraphs of importance for interconnection networks, Graph theory is also used by J.M. HAMMERSLEY in his study of the Friendship Theorem and the Love Problem. His paper looks back to classical mythology with references to Narcissus, but in producing it he has made use of the latest technology in the form of the Oxford University Laser-comp typesetting facility. Perhaps the day is not far off when it will become routine for authors to produce their papers by such means. Other papers using graph theory are those of Schrijver and Shult mentioned below.

INTRODUCTION

It is well known that telecommunication networks or interconnection networks can be modelled by graphs. Recent advances in technology, especially the advent of very large scale integrated (VLSI) circuit technology have enabled very complex interconnection networks to be constructed. Thus it is of great interest to study the topologies of interconnection networks, and, in particular, their associated graphical properties. If there are point-to-point connections, the computer network is modelled by a graph in which the nodes or vertices correspond to the computer centres in the network and the edges correspond to the communication links. When the computers share a communication medium such as a bus, the network is modelled by a hypergraph, where the nodes correspond to the computer centres and the (hyper)edges to the buses. Note that there exists a second important class of networks, the “multistage networks”, but we will not consider them. For a survey of interconnection networks, we refer the reader to Feng (1981).

In the design of these networks, several parameters are very important, for example message delay, message traffic density, reliability or fault tolerance, existence of efficient algorithms for routing messages, cost of the networks, …

one important measure of the power of an interconnection network is the length of the longest path that the messages must travel from one node to another in the network, i.e. the distance between the nodes. It is advantageous to make these distances as small as possible, since this will reduce the message delay and also the message traffic density in the links.

Community relations

In response to the editor's request for a survey article on a combinatorial topic, I have assembled under a single heading —the love problem, as I shall call it — material that has previously appeared in separate contexts and diverse guises, such as the solubility of Diophantine quadratic matrix equations, the construction of block designs, the existence of finite geometries, etc. However, I shall only mention a handful of references as leads into the extensive literature, which I could not hope to cover by anything approaching a complete bibliography. Moreover, in stressing the graph-theoretic aspects of the matter, I shall be adopting a rather different line from traditional treatments.

Let us begin with a special case of the love problem, known as the friendship theorem. I do not know who first stated this theorem: the earliest published paper that I have come across is Wilf (1971), who cites an earlier unpublished account by Graham Higman in 1968.

Introduction. Let (Q,o) be a quasigroup of order n and define an n2 × 3 array A by (x,y,z) is a row of A if and only if x o y = z. As a consequence of the fact that the equations a o x = b and y o a = b are uniquely solvable for all a,b ε Q, if we run our fingers down any two columns of A we get each ordered pair belonging to Q × Q e×actly once. An n2 × 3 array with this property is called an orthogonal array and it doesn't take the wisdom of a saint to see that this construction can be reversed. That is, if A is any n2 × 3 orthogonal array (defined on a set Q) and we define a binary operation o on Q by x o y = z if and only if (x,y,z) is a row of A, then (Q,o) is a quasigroup. Hence we can think of a quasigroup as an n2 × 3 orthogonal array and conversely. Now given an n2 × 3 orthogonal array A there is the irresistable urge to permute the columns of A. One of the reasons for this urge is that the resulting n2 × 3 array is still an orthogonal array. (If running our fingers down any two columns of A gives every ordered pair of Q × Q exactly once, the same must be true (of course) if we rearrange the columns.)

INTRODUCTION

A yery famous theorem (associated with the names Hilbert, von Staudt, Veblen and Young) characterizes projective spaces of dimension greater than 2 as linear incidence systems satisfying a certain (variously named) axiom. By the term “characterization”, one means a complete classification in terms of division rings. This famous characterization theorem fully displays the spirit of synthetic geometry in that one obtains an exact and elaborate structure with many subspaces from a few simple axioms mentioning only points and lines.

More than three decades later F. Buekenhout and the author obtained a characterization of polar spaces of rank more than 2 in terms of a similar set of very simple axioms concerning only points and lines. But this time the characterization rested on a considerably more involved theory of Veldkamp and Tits, where, in effect, the really difficult work was done. Indeed Tits' work on polar spaces (as axiomatized by him) was a part of his monumental classification of buildings of spherical type of rank greater than 2. The Buekenhout-Shult polar space theorem could then be seen as a characterization of the buildings of types C and D in terms of axioms involving only two types of varieties of the building. The question was then raised (see [20]) whether similar axiomatically simple “point-line” characterizations could be obtained for all the buildings of spherical type of rank at least 3.